Abhishek Shetty

Ishaq Aden-Ali, Yeshwanth Cherapanamjeri, Abhishek Shetty et al. · eth-zurich

In statistical learning theory, determining the sample complexity of realizable binary classification for VC classes was a long-standing open problem. The results of Simon and Hanneke established sharp upper bounds in this setting. However, the reliance of their argument on the uniform convergence principle limits its applicability to more general learning settings such as multiclass classification. In this paper, we address this issue by providing optimal high probability risk bounds through a framework that surpasses the limitations of uniform convergence arguments. Our framework converts the leave-one-out error of permutation invariant predictors into high probability risk bounds. As an application, by adapting the one-inclusion graph algorithm of Haussler, Littlestone, and Warmuth, we propose an algorithm that achieves an optimal PAC bound for binary classification. Specifically, our result shows that certain aggregations of one-inclusion graph algorithms are optimal, addressing a variant of a classic question posed by Warmuth. We further instantiate our framework in three settings where uniform convergence is provably suboptimal. For multiclass classification, we prove an optimal risk bound that scales with the one-inclusion hypergraph density of the class, addressing the suboptimality of the analysis of Daniely and Shalev-Shwartz. For partial hypothesis classification, we determine the optimal sample complexity bound, resolving a question posed by Alon, Hanneke, Holzman, and Moran. For realizable bounded regression with absolute loss, we derive an optimal risk bound that relies on a modified version of the scale-sensitive dimension, refining the results of Bartlett and Long. Our rates surpass standard uniform convergence-based results due to the smaller complexity measure in our risk bound.

Constantinos Daskalakis, Noah Golowich, Nika Haghtalab et al.

A fundamental shortcoming of the concept of Nash equilibrium is its computational intractability: approximating Nash equilibria in normal-form games is PPAD-hard. In this paper, inspired by the ideas of smoothed analysis, we introduce a relaxed variant of Nash equilibrium called $σ$-smooth Nash equilibrium, for a smoothness parameter $σ$. In a $σ$-smooth Nash equilibrium, players only need to achieve utility at least as high as their best deviation to a $σ$-smooth strategy, which is a distribution that does not put too much mass (as parametrized by $σ$) on any fixed action. We distinguish two variants of $σ$-smooth Nash equilibria: strong $σ$-smooth Nash equilibria, in which players are required to play $σ$-smooth strategies under equilibrium play, and weak $σ$-smooth Nash equilibria, where there is no such requirement. We show that both weak and strong $σ$-smooth Nash equilibria have superior computational properties to Nash equilibria: when $σ$ as well as an approximation parameter $ε$ and the number of players are all constants, there is a constant-time randomized algorithm to find a weak $ε$-approximate $σ$-smooth Nash equilibrium in normal-form games. In the same parameter regime, there is a polynomial-time deterministic algorithm to find a strong $ε$-approximate $σ$-smooth Nash equilibrium in a normal-form game. These results stand in contrast to the optimal algorithm for computing $ε$-approximate Nash equilibria, which cannot run in faster than quasipolynomial-time. We complement our upper bounds by showing that when either $σ$ or $ε$ is an inverse polynomial, finding a weak $ε$-approximate $σ$-smooth Nash equilibria becomes computationally intractable.

Ishaq Aden-Ali, Yeshwanth Cherapanamjeri, Abhishek Shetty et al. · eth-zurich

The one-inclusion graph algorithm of Haussler, Littlestone, and Warmuth achieves an optimal in-expectation risk bound in the standard PAC classification setup. In one of the first COLT open problems, Warmuth conjectured that this prediction strategy always implies an optimal high probability bound on the risk, and hence is also an optimal PAC algorithm. We refute this conjecture in the strongest sense: for any practically interesting Vapnik-Chervonenkis class, we provide an in-expectation optimal one-inclusion graph algorithm whose high probability risk bound cannot go beyond that implied by Markov's inequality. Our construction of these poorly performing one-inclusion graph algorithms uses Varshamov-Tenengolts error correcting codes. Our negative result has several implications. First, it shows that the same poor high-probability performance is inherited by several recent prediction strategies based on generalizations of the one-inclusion graph algorithm. Second, our analysis shows yet another statistical problem that enjoys an estimator that is provably optimal in expectation via a leave-one-out argument, but fails in the high-probability regime. This discrepancy occurs despite the boundedness of the binary loss for which arguments based on concentration inequalities often provide sharp high probability risk bounds.

Surbhi Goel, Steve Hanneke, Shay Moran et al.

We study the problem of sequential prediction in the stochastic setting with an adversary that is allowed to inject clean-label adversarial (or out-of-distribution) examples. Algorithms designed to handle purely stochastic data tend to fail in the presence of such adversarial examples, often leading to erroneous predictions. This is undesirable in many high-stakes applications such as medical recommendations, where abstaining from predictions on adversarial examples is preferable to misclassification. On the other hand, assuming fully adversarial data leads to very pessimistic bounds that are often vacuous in practice. To capture this motivation, we propose a new model of sequential prediction that sits between the purely stochastic and fully adversarial settings by allowing the learner to abstain from making a prediction at no cost on adversarial examples. Assuming access to the marginal distribution on the non-adversarial examples, we design a learner whose error scales with the VC dimension (mirroring the stochastic setting) of the hypothesis class, as opposed to the Littlestone dimension which characterizes the fully adversarial setting. Furthermore, we design a learner for VC dimension~1 classes, which works even in the absence of access to the marginal distribution. Our key technical contribution is a novel measure for quantifying uncertainty for learning VC classes, which may be of independent interest.

Alankrita Bhatt, Nika Haghtalab, Abhishek Shetty

We initiate the study of smoothed analysis for the sequential probability assignment problem with contexts. We study information-theoretically optimal minmax rates as well as a framework for algorithmic reduction involving the maximum likelihood estimator oracle. Our approach establishes a general-purpose reduction from minimax rates for sequential probability assignment for smoothed adversaries to minimax rates for transductive learning. This leads to optimal (logarithmic) fast rates for parametric classes and classes with finite VC dimension. On the algorithmic front, we develop an algorithm that efficiently taps into the MLE oracle, for general classes of functions. We show that under general conditions this algorithmic approach yields sublinear regret.

Don Kurian Dennis, Abhishek Shetty, Anish Sevekari et al.

We study the problem of progressive ensemble distillation: Given a large, pretrained teacher model $g$, we seek to decompose the model into smaller, low-inference cost student models $f_i$, such that progressively evaluating additional models in this ensemble leads to improved predictions. The resulting ensemble allows for flexibly tuning accuracy vs. inference cost at runtime, which is useful for a number of applications in on-device inference. The method we propose, B-DISTIL , relies on an algorithmic procedure that uses function composition over intermediate activations to construct expressive ensembles with similar performance as $g$ , but with smaller student models. We demonstrate the effectiveness of B-DISTIL by decomposing pretrained models across standard image, speech, and sensor datasets. We also provide theoretical guarantees in terms of convergence and generalization.

Adam Block, Alexander Rakhlin, Abhishek Shetty

In order to circumvent statistical and computational hardness results in sequential decision-making, recent work has considered smoothed online learning, where the distribution of data at each time is assumed to have bounded likeliehood ratio with respect to a base measure when conditioned on the history. While previous works have demonstrated the benefits of smoothness, they have either assumed that the base measure is known to the learner or have presented computationally inefficient algorithms applying only in special cases. This work investigates the more general setting where the base measure is \emph{unknown} to the learner, focusing in particular on the performance of Empirical Risk Minimization (ERM) with square loss when the data are well-specified and smooth. We show that in this setting, ERM is able to achieve sublinear error whenever a class is learnable with iid data; in particular, ERM achieves error scaling as $\tilde O( \sqrt{\mathrm{comp}(\mathcal F)\cdot T} )$, where $\mathrm{comp}(\mathcal F)$ is the statistical complexity of learning $\mathcal F$ with iid data. In so doing, we prove a novel norm comparison bound for smoothed data that comprises the first sharp norm comparison for dependent data applying to arbitrary, nonlinear function classes. We complement these results with a lower bound indicating that our analysis of ERM is essentially tight, establishing a separation in the performance of ERM between smoothed and iid data.

Adam Block, Mark Bun, Rathin Desai et al.

Due to statistical lower bounds on the learnability of many function classes under privacy constraints, there has been recent interest in leveraging public data to improve the performance of private learning algorithms. In this model, algorithms must always guarantee differential privacy with respect to the private samples while also ensuring learning guarantees when the private data distribution is sufficiently close to that of the public data. Previous work has demonstrated that when sufficient public, unlabelled data is available, private learning can be made statistically tractable, but the resulting algorithms have all been computationally inefficient. In this work, we present the first computationally efficient, algorithms to provably leverage public data to learn privately whenever a function class is learnable non-privately, where our notion of computational efficiency is with respect to the number of calls to an optimization oracle for the function class. In addition to this general result, we provide specialized algorithms with improved sample complexities in the special cases when the function class is convex or when the task is binary classification.

Dhruv Rohatgi, Abhishek Shetty, Donya Saless et al.

Test-time algorithms that combine the generative power of language models with process verifiers that assess the quality of partial generations offer a promising lever for eliciting new reasoning capabilities, but the algorithmic design space and computational scaling properties of such approaches are still opaque, and their benefits are far from apparent when one accounts for the cost of learning a high-quality verifier. Our starting point is the observation that seemingly benign errors in a learned verifier can lead to catastrophic failures for standard decoding techniques due to error amplification during the course of generation. We then ask: can this be improved with more sophisticated decoding strategies? We introduce a new process-guided test-time sampling algorithm, VGB, which uses theoretically grounded backtracking to achieve provably better robustness to verifier errors. VGB interprets autoregressive generation as a random walk on a tree of partial generations, with transition probabilities guided by the process verifier and base model; crucially, backtracking occurs probabilistically. This process generalizes the seminal Sinclair-Jerrum random walk (Sinclair & Jerrum, 1989) from the literature on approximate counting and sampling in theoretical computer science, and a conceptual contribution of our work is to highlight parallels with this literature. Empirically, we demonstrate on both synthetic and real language modeling tasks that VGB outperforms baselines on a variety of metrics.

Yeshwanth Cherapanamjeri, Sumegha Garg, Nived Rajaraman et al.

We study the memory complexity of machine unlearning algorithms that provide strong data deletion guarantees to the users. Formally, consider an algorithm for a particular learning task that initially receives a training dataset. Then, after learning, it receives data deletion requests from a subset of users (of arbitrary size), and the goal of unlearning is to perform the task as if the learner never received the data of deleted users. In this paper, we ask how many bits of storage are needed to be able to delete certain training samples at a later time. We focus on the task of realizability testing, where the goal is to check whether the remaining training samples are realizable within a given hypothesis class \(\mathcal{H}\). Toward that end, we first provide a negative result showing that the VC dimension is not a characterization of the space complexity of unlearning. In particular, we provide a hypothesis class with constant VC dimension (and Littlestone dimension), but for which any unlearning algorithm for realizability testing needs to store \(Ω(n)\)-bits, where \(n\) denotes the size of the initial training dataset. In fact, we provide a stronger separation by showing that for any hypothesis class \(\mathcal{H}\), the amount of information that the learner needs to store, so as to perform unlearning later, is lower bounded by the \textit{eluder dimension} of \(\mathcal{H}\), a combinatorial notion always larger than the VC dimension. We complement the lower bound with an upper bound in terms of the star number of the underlying hypothesis class, albeit in a stronger ticketed-memory model proposed by Ghazi et al. (2023). Since the star number for a hypothesis class is never larger than its Eluder dimension, our work highlights a fundamental separation between central and ticketed memory models for machine unlearning.

Omar Montasser, Abhishek Shetty, Nikita Zhivotovskiy

We revisit online binary classification by shifting the focus from competing with the best-in-class binary loss to competing against relaxed benchmarks that capture smoothed notions of optimality. Instead of measuring regret relative to the exact minimal binary error -- a standard approach that leads to worst-case bounds tied to the Littlestone dimension -- we consider comparing with predictors that are robust to small input perturbations, perform well under Gaussian smoothing, or maintain a prescribed output margin. Previous examples of this were primarily limited to the hinge loss. Our algorithms achieve regret guarantees that depend only on the VC dimension and the complexity of the instance space (e.g., metric entropy), and notably, they incur only an $O(\log(1/γ))$ dependence on the generalized margin $γ$. This stands in contrast to most existing regret bounds, which typically exhibit a polynomial dependence on $1/γ$. We complement this with matching lower bounds. Our analysis connects recent ideas from adversarial robustness and smoothed online learning.

Adam Block, Abhishek Shetty

In order to develop practical and efficient algorithms while circumventing overly pessimistic computational lower bounds, recent work has been interested in developing oracle-efficient algorithms in a variety of learning settings. Two such settings of particular interest are online and differentially private learning. While seemingly different, these two fields are fundamentally connected by the requirement that successful algorithms in each case satisfy stability guarantees; in particular, recent work has demonstrated that algorithms for online learning whose performance adapts to beneficial problem instances, attaining the so-called small-loss bounds, require a form of stability similar to that of differential privacy. In this work, we identify the crucial role that separation plays in allowing oracle-efficient algorithms to achieve this strong stability. Our notion, which we term $ρ$-separation, generalizes and unifies several previous approaches to enforcing this strong stability, including the existence of small-separator sets and the recent notion of $γ$-approximability. We present an oracle-efficient algorithm that is capable of achieving small-loss bounds with improved rates in greater generality than previous work, as well as a variant for differentially private learning that attains optimal rates, again under our separation condition. In so doing, we prove a new stability result for minimizers of a Gaussian process that strengthens and generalizes previous work.

Surbhi Goel, Abhishek Shetty, Konstantinos Stavropoulos et al.

We study the problem of learning under arbitrary distribution shift, where the learner is trained on a labeled set from one distribution but evaluated on a different, potentially adversarially generated test distribution. We focus on two frameworks: PQ learning [Goldwasser, A. Kalai, Y. Kalai, Montasser NeurIPS 2020], allowing abstention on adversarially generated parts of the test distribution, and TDS learning [Klivans, Stavropoulos, Vasilyan COLT 2024], permitting abstention on the entire test distribution if distribution shift is detected. All prior known algorithms either rely on learning primitives that are computationally hard even for simple function classes, or end up abstaining entirely even in the presence of a tiny amount of distribution shift. We address both these challenges for natural function classes, including intersections of halfspaces and decision trees, and standard training distributions, including Gaussians. For PQ learning, we give efficient learning algorithms, while for TDS learning, our algorithms can tolerate moderate amounts of distribution shift. At the core of our approach is an improved analysis of spectral outlier-removal techniques from learning with nasty noise. Our analysis can (1) handle arbitrarily large fraction of outliers, which is crucial for handling arbitrary distribution shifts, and (2) obtain stronger bounds on polynomial moments of the distribution after outlier removal, yielding new insights into polynomial regression under distribution shifts. Lastly, our techniques lead to novel results for tolerant testable learning [Rubinfeld and Vasilyan STOC 2023], and learning with nasty noise.

Nika Haghtalab, Yanjun Han, Abhishek Shetty et al.

In this paper, we study oracle-efficient algorithms for beyond worst-case analysis of online learning. We focus on two settings. First, the smoothed analysis setting of [RST11,HRS22] where an adversary is constrained to generating samples from distributions whose density is upper bounded by $1/σ$ times the uniform density. Second, the setting of $K$-hint transductive learning, where the learner is given access to $K$ hints per time step that are guaranteed to include the true instance. We give the first known oracle-efficient algorithms for both settings that depend only on the pseudo (or VC) dimension of the class and parameters $σ$ and $K$ that capture the power of the adversary. In particular, we achieve oracle-efficient regret bounds of $ \widetilde{O} ( \sqrt{T dσ^{-1}} ) $ and $ \widetilde{O} ( \sqrt{T dK} ) $ for learning real-valued functions and $ O ( \sqrt{T dσ^{-\frac{1}{2}} } )$ for learning binary-valued functions. For the smoothed analysis setting, our results give the first oracle-efficient algorithm for online learning with smoothed adversaries [HRS22]. This contrasts the computational separation between online learning with worst-case adversaries and offline learning established by [HK16]. Our algorithms also achieve improved bounds for worst-case setting with small domains. In particular, we give an oracle-efficient algorithm with regret of $O ( \sqrt{T(d |\mathcal{X}|)^{1/2} })$, which is a refinement of the earlier $O ( \sqrt{T|\mathcal{X}|})$ bound by [DS16].

Nika Haghtalab, Tim Roughgarden, Abhishek Shetty

We prove novel algorithmic guarantees for several online problems in the smoothed analysis model. In this model, at each time an adversary chooses an input distribution with density function bounded above by $\tfrac{1}σ$ times that of the uniform distribution; nature then samples an input from this distribution. Crucially, our results hold for {\em adaptive} adversaries that can choose an input distribution based on the decisions of the algorithm and the realizations of the inputs in the previous time steps. This paper presents a general technique for proving smoothed algorithmic guarantees against adaptive adversaries, in effect reducing the setting of adaptive adversaries to the simpler case of oblivious adversaries. We apply this technique to prove strong smoothed guarantees for three problems: -Online learning: We consider the online prediction problem, where instances are generated from an adaptive sequence of $σ$-smooth distributions and the hypothesis class has VC dimension $d$. We bound the regret by $\tilde{O}\big(\sqrt{T d\ln(1/σ)} + d\sqrt{\ln(T/σ)}\big)$. This answers open questions of [RST11,Hag18]. -Online discrepancy minimization: We consider the online Komlós problem, where the input is generated from an adaptive sequence of $σ$-smooth and isotropic distributions on the $\ell_2$ unit ball. We bound the $\ell_\infty$ norm of the discrepancy vector by $\tilde{O}\big(\ln^2\!\big( \frac{nT}σ\big) \big)$. -Dispersion in online optimization: We consider online optimization of piecewise Lipschitz functions where functions with $\ell$ discontinuities are chosen by a smoothed adaptive adversary and show that the resulting sequence is $\big( σ/{\sqrt{T\ell}}, \tilde O\big(\sqrt{T\ell} \big)\big)$-dispersed. This matches the parameters of [BDV18] for oblivious adversaries, up to log factors.